可算函数

Several strong results on neural network approximation capability obtained in this dissertation are important in the approximation theory and applications of neural networks.

本文系统地研究了多层神经网络的非线性逼近能力，给出了多层网络可一致逼近有限维空间R〓紧集上的连续函数、无穷维函数空间紧集上的连续泛函和连续算子以及布尔函数的理论证明。

Firstly, we get the necessary and sufficient conditions for Toeplitz operators that commutes with another such operator whose symbol is a monomial, meanwhile, we completely characterize when the mellin transform of the bounded function on the interval 0,1 is a rational function.

首先得到了以有界函数为符号的Toeplitz算子和以单项式函数为符号的Toeplitz算子可交换的充要条件，同时给出了0，1区间上的有界函数的Mellin变换为有理函数的充要条件。

Thirdly,quasi-normal inequalities of a-variation maximal operator and a-conditional variation maximal operator of scalar predictable tree martingales areidentified by the use of martingale transforms and by the construction of convex or concave function method;on this basis and with the help of previsiblity or regu-larity,Burkholder-Davis-Gundy\'s inequality of a-variation maximal operator and a-conditional variation maximal operator of scalar predictable tree martingales are iden-tified by the application of Hardy-Lorentz interpolation theory.At the same time,bythe use of G.

再次，应用鞅变换和构造凸或凹函数方法证明了标量值可料树鞅的α-方极大算子和α-条件方极大算子的拟范数不等式；然后，在这些拟范数不等式的基础上，应用Hardy-Lorentz空间插值方法证明了当树鞅是可控或正规树鞅时关于标量值可料树鞅α-方极大算子和α-条件方极大算子的Burkholder-Davis-Gundy's不等式成立。

On the basis of general Krawczyk-Hansen Operator, the interval iteration of the constrained functions are established; the interval extension of maximum entropy function is discussed; convergence is proved; region deletion rules are supplied and the algorithm of interval maximum entropy has been set up.

利用极大熵函数和罚函数将问题转化为无约束可微优化问题，借助广义Krawczyk-Hansen算子建立了约束函数的区间迭代；讨论了极大熵函数和罚函数的区间扩张，证明了收敛性等性质，给出了无解区域删除原则，建立了区间极大熵算法。

However, the study, as a part of linear operator theory, of operators induced by composition with a fixed function has a relatively short history.

但是一个固定的函数与某个函数空间上的函数复合作为该空间上的线性算子进行研究则是不久的事情，这可追溯到二十世纪60年代末E。

This dissertation proves that the class of monotone increase functions is PAC solvable while the class of single peak functions is non-PAC solvable under a GA with reproduction and a mutation operator which mutate according to a pre-set order.

本文提出了概率意义上近似收敛可解的概念，并证明了随机行走和狄拉克δ函数都是非PAC可解的，证明了任何问题在只有杂交算符或变异算符的遗传算法下都是非PAC可解的。

Some conclusions are obtained:(1) The interval data concept can rationally settle the apart component's dependence problem of monitoring data;(2) The survey point's displacement which is the material parameter's increasing function or deducing function can be used to back-analyse the material parameter's up and down limit; and (3) Both the interval parameter perturbation back analysis method and the interval parameter monotony back analysis method can obtain satisfactory interval parameters.

通过分析得出结论：(1)可利用区间数概念合理解决监测资料中分离分量相关性的问题；(2)可利用工程结构中位移为待反分析材料参数的增函数或减函数的测点位移的上、下限，来反演得到不确定性材料参数的上、下限；(3)工程算例分析表明，区间参数摄动优化反分析法和区间参数单调性优化反分析法都能得到令人满意的参数区间。

The fourth chapter analysesthe differences of the resistance between the boundariescalculated by different models.It can seen from the results thatthe differences between resistances calculated by variousfunctions can ignored under unstable conditions.There areobvious differences between different resistances calculated bydifferent functions under stable conditions.However,in bothcases,error from any function to predict transpiration rate isignoble.In order to make more applicable,the Monteith modelwas fished in this chapter.

第四章分析了各种边界层阻力计算模型计算结果的差异，比较结果表明，在空气层结不稳定时，各种φm函数经验模型计算阻力的差异可忽略，而当空气层结稳定时各种φm函数经验模型所算边界层阻力有明显差异，但不论是稳定和非稳定层结下，任选一中φm函数计算模型对蒸腾速率计算结果所造成的差异可忽略，为便于应用，本章还比较和检验了Monteith模型计算结果的差异，结果表明，该模型用于蒸腾计算，其误差可忽略不计。

From our results we know that the average error of the Lagrange interpolation sequence and the Hermite interpolation sequence based on the Chebyshev nodes in the 1-fold integrated Wiener space equal weakly to the average error of their corresponding optimal approximation polynomial in the 1-fold integrated Wiener space,and as a kind of information-based operation,they have simple form and their recover functions are polynomials,in the 1-fold integrated wiener space,their average error equal weakly to the corresponding minimal information radius whose permissible information operators class is function values.

通过我们的结果可以知道，基于第一类Chebyshev多项式零点的Lagrange插值算子列和Hermite插值算子列在1-重积分Wiener空间下的平均误差弱等价于相应的最佳逼近多项式在1-重积分Wiener空间下的平均误差，并且作为形式简单且恢复函数为多项式的一种信息基算法，其在1-重积分Wiener空间下的平均误差弱等价于相应的以函数值计算为可允许信息算子的最小平均信息半径。

This paper presents a one- way trapdoor function based on the favorable properties of ergodic matrices, then implements their applications in secret- key cryptography, hybrid cryptography, STS Protocol and Shamir Three-Pass Protocol.

单单研究单向函数是没有意义的，没有什么方法能够算出它，但是如果在单向函数中加一个条件，使得原先不可解的函数变为可解，那么对于加密就有重要的意义。

On the other hand, the more important thing is because the urban housing is a kind of heterogeneity products.

另一方面，更重要的是由于城市住房是一种异质性产品。

Climate histogram is the fall that collects place measure calm value, cent serves as cross axle for a few equal interval, the area that the frequency that the value appears according to place is accumulated and becomes will be determined inside each interval, discharge the graph that rise with post, also be called histogram.

气候直方图是将所收集的降水量测定值，分为几个相等的区间作为横轴，并将各区间内所测定值依所出现的次数累积而成的面积，用柱子排起来的图形，也叫做柱状图。